Second, the statement of Theorem 1.1 from [1]} replaces Item REF  with the weaker requirement that \(\mathcal {E}_{\delta }(A-A) \le K_3 \mathcal {E}_{\delta }(A),\)  and replaces Item REF  with the stronger requirement that \(\mathcal {E}_{\delta }(A.A)\le K_4\mathcal {E}_{\delta }(A)\) . The authors in [1]} begin by finding a large subset \(A^{\prime }\subset A\)  that satisfies Item REF  (in the form stated above), and then use their upper bound on \(\mathcal {E}_{\delta }(A.A)\)  (which also holds with \(A^{\prime }\)  in place of \(A\) ) to obtain Item REF , with \(A^{\prime }\)  in place of \(A\) . In our setting, we are not allowed to replace \(A\)  with a subset \(A^{\prime }\subset A\) , since Item REF  might fail for \(A^{\prime }\) . This is why we impose the stronger condition in Item REF  at the outset. we will sketch a proof of Proposition REF  in Appendix .
